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A symmetry problem in polymer calculations
Volume 25; number 2
CHEMICAL PHYSICS LETTERS
15 March 1974
A SYMMETRY PROBLEM IN POLYMER CALCULATIONS Searnus F, O’SHEA*
and David P. SANTRY
Department of Chemistry. h&Master University. Numifton. Ontario. Chada Received 10 September 1973 Revised manuscript received 18 December 1973
It is shown by a symmetry argument that the band gap in a symmetric polyene must be zero in the restricted single determinant approximation. Spurious band gaps can arise in SCF calculations when the symmetry of the lattice is not refixted in the lattice sums. The consequences of this symmetry probIem are discussed.
Advances in computational equipment have Ied to a marked increase in the tlumber of calculations of the electronic structures of polymers and solids. They
have also led to the use of more realistic approximations in such calculations. The incksion of nonnearest neighbour interactions in the evaluation of matrix elements of the various quantum mechanical operators is expected to lead to significant improvement in calculated properties. However, the inclusion of these interactions can lead to serious quantitative errors, and even qualitative errors, when it is not done correctly. This is illustrated here by reference to calculations of the electronic structure of symmetric polyenes.
A number of studies of the electronic structure of infinite polyenes, both symmetric (all carbon-carbon bond lengths equal) and altemant (alternate C-C bonds long and short) have recently been published [1,2]. One of the quantities of interest is the band gap between the x and II* levels for these structures. In a previous publication [I 1 the present authors implied that the band gap for the symmetric polyene is very small but not zero; ref. [2] also reports a non-zero value for the band gap of the symmetric polyene. We show here that this gap must equal zero for the symmetric pol$ene and that a spurious non-zero value * Department of Chemistry, University of Western Ontario; London, Ontario, Canada.
can result from an imbalance in the lattice sums used for the calculation. That the R and T* orbitals must be degenerate at the edge of the Brillouin zone can easily be verified by a direct examination of the orbitals involved [3].
A more direct proof follows by showing that the molecular orbital @,Jk = 112) transforms into @,&c = 112) on application of an element from the one-dimensional space group. Let 01, fig. 1, be a reflection plane from the Ith unit celi and $ and $ bit& from the rth cell, then
the vr atomic or-
Without loss of generality we can assume I labels the reference.cell, and thus is equal to zero. The crystal orbitats are given by
+(k)
=
Therefore
xr exp@rikr)xf+
c t :
exp[2nik(r+l/2)]$
_
15 March 1974
CHEMICAL PHYSICS LElTERS
Volume 25. number 2
band gap, table i. This spurious gap decreases as the number of the terms in thesummation is increased, but is still appreciable when 37 unit cells are included. We thus conclude that the a and zr*.bands touch at the edge of the Brillouin zone for the symmetric polymer and that an unrestricted Hartree-Fock approach, or its equivalent, is required for calculations on this structure. Also, spurious band gaps can appear unless care is taken to balance the slowly convergent lattice sums which are used in the calculation. out balancing the sums, it is impossible
Fig. 1. Unit cell from the symmetric polyene. The shorter broken lines are the unit cell boundaries. If the cell shown were at the origin, then, in the notation of the text, the reflection plant would be OT_ Thus, at k = l/2, & and @,+ are the two components from a degenerate representation and must therefore have the same orbital energy. An apparent band gap can result for the symmetric polyene if the various lattice sums used in the calculation are not properly balanced to reflect the true lattice symmetry. That is, the number of terms, included in the lattice sum from the positive direction must exactly equal the number from the negative direction. It is important to realise that, depending on the structure of the unit cell, this does nor necessarily imply that the number of unit cells must be equal. We illustrate this effect with some simple calculations based on the CNDO/2 approximation, table 1. These unbalanced results are based on lattice sums which include an equal number of unit cells in both directions. Fig. 1 shows that such an arrangement actually leads to an imbalance in the I-II type of lattice sum. This imbalance is reflected by the appearance of a spurious
indeed, with-
to get proper convergence for many of the calculation properties. The above analysis applies to calculations for any polymer system. In the particular case of the polyene a number of additional comments are in order. The work of Paldus and Cizek [4] has demonstrated that Hartree-Fock wavefunctions for large cyclic polyenes are unstable with respect to singlet instabilities. That the Hartree-Fock wavefunction for infinite llnear polyene is likewise unstable is easily demonstrated: when the SCF procedure for the symmetric polyene is initiated using as a first guess the wavefunciion for a neighbouring altemant configuration a stable converged solution with altemant symmetry and a de& nite band gap results. In a case such as this, where the normal molecular orbital hamiltonian operator is inadequate, the above analysis is also inadequate. In view of this the infinite polyene is not a suitable system on which to test restricted single determinant theories. The implications of these results for the general problem are clear. Lattice summations must be performed in such a way that the symmetry of the lattice is preserved or else continued until the contrlbuting matrix elements in the direct lattice [3] become vanishingly small. If this prescription is not followed, the results will be quantitatively in error and possibly,
Table 1 me effect of truncation of lattice sums in the polyene calculation_ NC is the number of unit cells included in the Coulomb lattice sums. The terms “balanced” and “unb&nced” are discussed in the text. The calculations quoted here were done in the CNDO ap_ proxirmation as described in ref. [l] NC=5
Ecell
tau)
band JFP WI
ivc = 37
balanced
unbalanced
balanced
-15.6693
-15.6725 3.7
-15.6719 0.0
0.0
unbalanced
’
-15.6717 0.27
165
Voluine.25, number 2
..
CHEMICAL PHYSICS LETTERS
..in cases such as the polyene band gap, qualitatively
in error_ When a system has little or no sym&etry, such as a polypeptide or an amorphous solid, the only unambiguous procedure is to continue the lattice sums until the matrix elements in the direct lattice have converged to -zero.
The authors gratefully acknowledge the financial support of the National Research Council of Canada. Helpful discussions with Drs. J_ Cizek and J. Paldus
are also gratefully acknowledged.
Z5 March 1974
References [I] S-F. O’Shea and D.P. Santry, J. Chem. Phys. 54 (1971) 2667. [2] J.M. Andre’and G. Leroy, Intern. J. Quantum Chem. 5 (1971) 557. j3] S-F. O’Shea, Ph.D. Thesis, M&faster University (1973). [4] J. Paldus and J. Cizek, Phys. Rev. A2 (1970) 2268, and references therein.
CHEMICAL PHYSICS LETTERS
15 March 1974
A SYMMETRY PROBLEM IN POLYMER CALCULATIONS Searnus F, O’SHEA*
and David P. SANTRY
Department of Chemistry. h&Master University. Numifton. Ontario. Chada Received 10 September 1973 Revised manuscript received 18 December 1973
It is shown by a symmetry argument that the band gap in a symmetric polyene must be zero in the restricted single determinant approximation. Spurious band gaps can arise in SCF calculations when the symmetry of the lattice is not refixted in the lattice sums. The consequences of this symmetry probIem are discussed.
Advances in computational equipment have Ied to a marked increase in the tlumber of calculations of the electronic structures of polymers and solids. They
have also led to the use of more realistic approximations in such calculations. The incksion of nonnearest neighbour interactions in the evaluation of matrix elements of the various quantum mechanical operators is expected to lead to significant improvement in calculated properties. However, the inclusion of these interactions can lead to serious quantitative errors, and even qualitative errors, when it is not done correctly. This is illustrated here by reference to calculations of the electronic structure of symmetric polyenes.
A number of studies of the electronic structure of infinite polyenes, both symmetric (all carbon-carbon bond lengths equal) and altemant (alternate C-C bonds long and short) have recently been published [1,2]. One of the quantities of interest is the band gap between the x and II* levels for these structures. In a previous publication [I 1 the present authors implied that the band gap for the symmetric polyene is very small but not zero; ref. [2] also reports a non-zero value for the band gap of the symmetric polyene. We show here that this gap must equal zero for the symmetric pol$ene and that a spurious non-zero value * Department of Chemistry, University of Western Ontario; London, Ontario, Canada.
can result from an imbalance in the lattice sums used for the calculation. That the R and T* orbitals must be degenerate at the edge of the Brillouin zone can easily be verified by a direct examination of the orbitals involved [3].
A more direct proof follows by showing that the molecular orbital @,Jk = 112) transforms into @,&c = 112) on application of an element from the one-dimensional space group. Let 01, fig. 1, be a reflection plane from the Ith unit celi and $ and $ bit& from the rth cell, then
the vr atomic or-
Without loss of generality we can assume I labels the reference.cell, and thus is equal to zero. The crystal orbitats are given by
+(k)
=
Therefore
xr exp@rikr)xf+
c t :
exp[2nik(r+l/2)]$
_
15 March 1974
CHEMICAL PHYSICS LElTERS
Volume 25. number 2
band gap, table i. This spurious gap decreases as the number of the terms in thesummation is increased, but is still appreciable when 37 unit cells are included. We thus conclude that the a and zr*.bands touch at the edge of the Brillouin zone for the symmetric polymer and that an unrestricted Hartree-Fock approach, or its equivalent, is required for calculations on this structure. Also, spurious band gaps can appear unless care is taken to balance the slowly convergent lattice sums which are used in the calculation. out balancing the sums, it is impossible
Fig. 1. Unit cell from the symmetric polyene. The shorter broken lines are the unit cell boundaries. If the cell shown were at the origin, then, in the notation of the text, the reflection plant would be OT_ Thus, at k = l/2, & and @,+ are the two components from a degenerate representation and must therefore have the same orbital energy. An apparent band gap can result for the symmetric polyene if the various lattice sums used in the calculation are not properly balanced to reflect the true lattice symmetry. That is, the number of terms, included in the lattice sum from the positive direction must exactly equal the number from the negative direction. It is important to realise that, depending on the structure of the unit cell, this does nor necessarily imply that the number of unit cells must be equal. We illustrate this effect with some simple calculations based on the CNDO/2 approximation, table 1. These unbalanced results are based on lattice sums which include an equal number of unit cells in both directions. Fig. 1 shows that such an arrangement actually leads to an imbalance in the I-II type of lattice sum. This imbalance is reflected by the appearance of a spurious
indeed, with-
to get proper convergence for many of the calculation properties. The above analysis applies to calculations for any polymer system. In the particular case of the polyene a number of additional comments are in order. The work of Paldus and Cizek [4] has demonstrated that Hartree-Fock wavefunctions for large cyclic polyenes are unstable with respect to singlet instabilities. That the Hartree-Fock wavefunction for infinite llnear polyene is likewise unstable is easily demonstrated: when the SCF procedure for the symmetric polyene is initiated using as a first guess the wavefunciion for a neighbouring altemant configuration a stable converged solution with altemant symmetry and a de& nite band gap results. In a case such as this, where the normal molecular orbital hamiltonian operator is inadequate, the above analysis is also inadequate. In view of this the infinite polyene is not a suitable system on which to test restricted single determinant theories. The implications of these results for the general problem are clear. Lattice summations must be performed in such a way that the symmetry of the lattice is preserved or else continued until the contrlbuting matrix elements in the direct lattice [3] become vanishingly small. If this prescription is not followed, the results will be quantitatively in error and possibly,
Table 1 me effect of truncation of lattice sums in the polyene calculation_ NC is the number of unit cells included in the Coulomb lattice sums. The terms “balanced” and “unb&nced” are discussed in the text. The calculations quoted here were done in the CNDO ap_ proxirmation as described in ref. [l] NC=5
Ecell
tau)
band JFP WI
ivc = 37
balanced
unbalanced
balanced
-15.6693
-15.6725 3.7
-15.6719 0.0
0.0
unbalanced
’
-15.6717 0.27
165
Voluine.25, number 2
..
CHEMICAL PHYSICS LETTERS
..in cases such as the polyene band gap, qualitatively
in error_ When a system has little or no sym&etry, such as a polypeptide or an amorphous solid, the only unambiguous procedure is to continue the lattice sums until the matrix elements in the direct lattice have converged to -zero.
The authors gratefully acknowledge the financial support of the National Research Council of Canada. Helpful discussions with Drs. J_ Cizek and J. Paldus
are also gratefully acknowledged.
Z5 March 1974
References [I] S-F. O’Shea and D.P. Santry, J. Chem. Phys. 54 (1971) 2667. [2] J.M. Andre’and G. Leroy, Intern. J. Quantum Chem. 5 (1971) 557. j3] S-F. O’Shea, Ph.D. Thesis, M&faster University (1973). [4] J. Paldus and J. Cizek, Phys. Rev. A2 (1970) 2268, and references therein.